Algebra

TBA.

TBA.

TBA.

TBA.

TBA.

TBA.

TBA.

I shall explain Theorem (127) which determines the unique bijection BETWEEN the monomial basis called $(A_n)$- FC elements  AND the set of non-crossing diagrams of $n+1$ strings OF  our well-known Temperley-Lieb algebra, that respects the  diagrammatic multiplication by concatenation along with the two algorithms implementing this bijection and its inverse, in other terms : "Drawing" a basis element into a diagram and "writing" a diagram as member in the monomial basis.

Naturally  we shall find ourselves in CataLand, so I will try to give -as much as our time allows me- some consequences and open problems coming from the above holly marriage, that is to explain why did I commit this work (other than the obvious reasons). The talk is pretty simple & basically addressed to our Ph.D students, accessible to Master students and in which there is an introduction to my talk next week.

A gentle quiver is the data of a finite connected directed graph together with a collection of paths of length two satisfying additional conditions. A resolving subcategory of its representations is an additive subcategory that contains the projective objects and is closed by extensions and epimorphism kernels. In our framework, such a subcategory can be described combinatorially via a collection of indecomposable representations stable under some computational conditions.

In this algebraic context, a goal is to describe all resolving subcategories. To this end, we restrict ourselves to gentle trees (the directed graph is a tree) and use a geometric model to see indecomposable representations as curves on a disk. We then construct an algorithm that will enable us to compute them explicitly.

After reviewing all the essential notions and giving some motivations to understand the context, I will explain how we first describe the monogeneous resolving subcategories (generated by a single indecomposable nonprojective representation). Then, I will give some words on the design of the algorithm that allows the construction of all the resolving subcategories of any gentle tree. If time allows, I will share some expectations we can have following those results (link with tilting representations, generalization to gentle quivers, graduated cases, etc...) — all of this with combinatorial and geometrical perspectives.

This is a joint work in progress with Michael Schoonheere.

Stability conditions are an important tool in algebraic geometry for constructing moduli varieties. When applied to the varieties of modules over a finite-dimensional algebra, they give rise to the algebraic notion of semistable modules, which are closely linked to $tau$-tilting theory and cluster algebras. To find these semistable modules, one can compute a special class of regular functions known as determinantal semi-invariants. In this talk, we will revisit the relation of these semi-invariants to projective presentations and explore semistability for varieties of projective presentations. We will recall that determinantal semi-invariants give rise to two interesting types of subcategories, namely, wide subcategories of the module category and thick subcategories of the extriangulated category of projective presentations. Finally, we will introduce an extriangulated version of the correspondences among support $tau$-tilting objects, torsion classes, and wide subcategories. This correspondence extends classical results to the context of projective presentations.